All Questions
6
questions
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Is it possible to find a closed form for $i!$? [duplicate]
I am curious is there a closed form for $i!$? I tried to search for any closed form for this but I didn't find any.
$$z! := \lim_{n \to \infty } n^z \prod_{k=1}^n \frac {k}{z+k}$$
$$i! =\lim_{n \to \...
3
votes
2
answers
191
views
Proving $\sum_{n=-\infty}^\infty n^2e^{-\pi n^2}=\frac{\Gamma (1/4)}{4\sqrt{2}\pi^{7/4}}$
I conjecture that
$$\sum_{n=-\infty}^\infty n^2e^{-\pi n^2}=\frac{\Gamma (1/4)}{4\sqrt{2}\pi^{7/4}}$$
because the left-hand side and right-hand side agree to at least $50$ decimal places. Is the ...
- 667
6
votes
1
answer
307
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Closed form of $\sum_{n=1}^\infty \frac{1}{\sinh n\pi}$ in terms of $\Gamma (a)$, $a\in\mathbb{Q}$
This question and this question are about
$$\sum_{n=1}^\infty \frac{1}{\cosh n\pi}=\frac{1}{2}\left(\frac{\sqrt{\pi}}{\Gamma ^2(3/4)}-1\right)$$
and
$$\sum_{n=1}^\infty \frac{1}{\sinh ^2n\pi}=\frac{1}{...
- 966
3
votes
1
answer
149
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Closed form for $\Gamma (a+bi)\Gamma(a-bi)$ [duplicate]
I noticed that
$$\Gamma (3+2i)\Gamma (3-2i)=\frac{160\pi}{e^{2\pi}-e^{-2\pi}}$$
and
$$\Gamma (2+5i)\Gamma (2-5i)=\frac{260\pi}{e^{5\pi}-e^{-5\pi}}.$$
Is there a closed form for $\Gamma (a+bi)\Gamma (a-...
- 311
1
vote
1
answer
90
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How do you calculate the case $\lambda=2$ of this identity related to $\int_0^\infty \frac{x^{\lambda(s-1)}}{e^x+1}dx$?
Inspired in an integral representation for the Dirichlet Eta function (that is the alternating series of the Riemann Zeta function) I've calculated some integrals using Wolfram Alpha.
Example 1. ...
user243301
2
votes
1
answer
143
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Proof that $\int_0^1\frac{(-\log u)^s}{u^s}du=\frac{\Gamma(s+1)}{(1-s)^{s+1}}$ for $|\Re s|<1$
After I've read an identity involving an integral related with special functions, I've consider a different integral by trials asking to Wolfram Alpha online calculator
Example For the code int_0^1 ...
user243301